GRADE: Gibbs reaction and diffusion equations

By Song-Chun Zhu, David Mumford. 1998

Recently there have been increasing interests in using nonlinear PDEs for applications in computer vision and image processing. In this paper, we propose a general statistical framework for designing a new class of PDEs. For a given application, a Markov random field model p(I) is learned according to the minimax entropy principle so that p(I) should characterize the ensemble of images in our application. P(I) is a Gibbs distribution whose energy terms can be divided into two categories. Subsequently the partial differential equations given by gradient descent on the Gibbs potential are essentially reaction-diffusion equations, where the energy terms in one category produce anisotropic diffusion while the inverted energy terms in the second category produce reaction associated with pattern formation. We call this new class of PDEs the Gibbs Reaction And Diffusion Equations-GRADE and we demonstrate experiments where GRADE are used for texture pattern formation, denoising, image enhancement, and clutter removal.

Subfields: Computer Vision and Pattern Recognition, Probabilistic & Bayesian Methods

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computer Vision and Pattern Recognition

1. Song-Chun Zhu
2. Ying Nian Wu
3. Erik Nijkamp
4. Mitch Hill
5. Tian Han
6. Jianwen Xie
7. Luc Van Gool
8. Ruiqi Gao

In Probabilistic & Bayesian Methods

1. Song-Chun Zhu
2. Ying Nian Wu
3. Erik Nijkamp
4. Mitch Hill
5. Arthur Gretton
6. Yang Lu
7. Jianwen Xie
8. Tian Han

Citations in the corpus

• Learning Energy-Based Spatial-Temporal Generative ConvNets for Dynamic Patterns (2019, methodology): “The FRAME (Filters, Random fields, And Maximum Entropy) model [31], [32], [33] is a Markov”
• Learning Non-Convergent Non-Persistent Short-Run MCMC Toward Energy-Based Model (2019, methodology): “An early version of EBM is the FRAME (Filters, Random field, And Maximum Entropy) model [51, 45, 50], which is an exponential family model, where the features are the responses from a bank of filters.”
• Learning Generative ConvNets via Multi-grid Modeling and Sampling (2017, methodology): “The expectation in equation (5) is analytically intractable and has to be approximated by MCMC, such as the Langevin dynamics [48, 7], which iterates the following step:”
• A Theory of Generative ConvNet (2016, background): “The deterministic part of the Langevin equation (19) defines an attractor dynamics that converges to a local energy minimum (Zhu & Mumford, 1998).”
• Learning FRAME Models Using CNN Filters (2015, methodology): “Such a dynamics was first applied to the FRAME model by Zhu and Mumford (1998), and the gradient descent part of the dynamics was interpreted as the Gibbs Reaction And Diffusion Equations (GRADE).”
• Divergence Triangle for Joint Training of Generator Model, Energy-Based Model, and Inferential Model (2018, methodology): “[8] used gradient-based method such as ar X iv :1 81 2.”
• Cooperative Training of Descriptor and Generator Networks (2016, methodology): “Sampling synthesized examples from the current model by Langevin dynamics [22], [23], [24].”
• On Learning Non-Convergent Short-Run MCMC Toward Energy-Based Model (2019, methodology): “An early version of EBM is the FRAME (Filters, Random field, And Maximum Entropy) model [56, 47, 55], which is an exponential family model, where the features are the responses from a bank of filters.”
• The Nonlinear Statistics of High-Contrast Patches in Natural Images (2003, background): “, 2001), image reconstruction (Nielsen and Lillholm, 2001), denoising (Zhu and Mumford, 1998; Simoncelli, 1999a) and compression (Buccigrossi and Simoncelli, 1999).”
• Learning Latent Space Energy-Based Prior Model (2020, methodology): “respectively, which is about the reconstruction error. Expectations in (7) and (8) require MCMC sampling of the prior model p (z) and the posterior distribution p (zjx). We can use Langevin dynamics [32, 65]. For a target distribution ˇ(z), the dynamics iterates z k+1 = z k+sr zlogˇ(z k)+ p 2s k; (9”
• Exponential Family Estimation via Adversarial Dynamics Embedding (2019, methodology): “The finite-step dynamics-based MCMC sampler can be used for approximating Epf (x) [rff (x)] in rfL (f), which leads to the CD algorithm (Hinton, 2002; Zhu and Mumford, 1998).”